Functional maps representation on product manifolds

We consider the tasks of representing, analysing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace–Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices

Non-Rigid Puzzles

Shape correspondence is a fundamental problem in computer graphics and vision, with applications in various problems including animation, texture mapping, robotic vision, medical imaging, archaeology and many more. In settings where the shapes are allowed to undergo non-rigid deformations and only partial views are available, the problem becomes very challenging. To this end, we present a non-rigid multi-part shape matching algorithm. We assume to be given a reference shape and its multiple parts undergoing a non-rigid deformation. Each of these query parts can be additionally contaminated by clutter, may overlap with other parts, and there might be missing parts or redundant ones. Our method simultaneously solves for the segmentation of the reference model, and for a dense correspondence to (subsets of) the parts. Experimental results on synthetic as well as real scans demonstrate the effectiveness of our method in dealing with this challenging matching scenario

Force platform recordings in the diagnosis of primary orthostatic tremor

Primary orthostatic tremor (OT) consists of rhythmical muscle contractions at a frequency of around 16 Hz, causing discomfort and/or unsteadiness while standing. Diagnosis has hitherto relied on recording Electromyography (EMG) from affected muscles. The main aim of this study was to see if the characteristic postural tremor in OT can be identified with force platforms. We also quantified postural sway in OT patients to assess their degree of objective unsteadiness. Finally, we investigated the time relations between bursts of activity in the various affected muscle groups. Subjects stood on a force platform with concurrent multichannel surface EMG recordings from the lower limbs. Seven patients with clinical and EMG diagnosis of OT were examined and the force platform data compared with those of 21 other neurological patients with postural tremor and eight normal controls. All OT patients had high frequency peaks in power spectra of posturography and EMG recordings (12–16 Hz). No such high frequency activity was evident in patients with Parkinson's disease, cerebellar degenerations, essential tremor or in healthy controls. Additionally, OT patients showed increased sway at low frequencies relative to normal controls, suggesting that the unsteadiness reported by OT patients is at least partly due to increased postural sway. Examination of EMG timing showed fixed patterns of muscle activation when maintaining a quiet stance within but not across OT patients. These data show a high correlation between EMG and posturography and confirm that OT may be diagnosed using short epochs of force platform recordings

Making Laplacians commute

In this paper, we construct multimodal spectral geometry by finding a pair of closest commuting operators (CCO) to a given pair of Laplacians. The CCOs are jointly diagonalizable and hence have the same eigenbasis. Our construction naturally extends classical data analysis tools based on spectral geometry, such as diffusion maps and spectral clustering. We provide several synthetic and real examples of applications in dimensionality reduction, shape analysis, and clustering, demonstrating that our method better captures the inherent structure of multi-modal data

Naturalness and stability of the generalized Chaplygin gas in the seesaw cosmon scenario

The seesaw mechanism is conceived on the basis that a mass scale, $\xi$, and a dimensionless scale, $s$, can be fine-tuned in order to control the dynamics of active and sterile neutrinos through cosmon-type equations of motion: the seesaw cosmon equations. This allows for sterile neutrinos to be a dark matter candidate. In this scenario, the dynamical masses and energy densities of active and sterile neutrinos can be consistently embedded into the generalized Chaplygin gas (GCG), the unified dark sector model. In addition, dark matter adiabatically coupled to dark energy allows for a natural decoupling of the (active) mass varying neutrino (MaVaN) component from the dark sector. Thus MaVaN's turn into a secondary effect. Through the scale parameters, $\xi$ and $s$, the proposed scenario allows for a convergence among three distinct frameworks: the cosmon scenario, the seesaw mechanism for mass generation and the GCG model. It is found that the equation of state of the perturbations is the very one of the GCG background cosmology so that all the results from this approach are maintained, being smoothly modified by active neutrinos. Constrained by the seesaw relations, it is shown that the mass varying mechanism is responsible for the stability against linear perturbations and is indirectly related to the late time cosmological acceleration.Comment: 24 pages, 6 figure